What is the Integrating Factor for x^n*y^m?

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SUMMARY

The discussion focuses on finding an integrating factor of the form xnym to solve the differential equation (12 + 5xy)dx + (6(x/y) + 3x2)dy = 0. Participants emphasize the necessity of transforming the equation into an exact form by ensuring the derivatives of the modified equation match. Specifically, the derivatives yield two equations that must be solved for the variables m and n, leading to the determination of the integrating factor.

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Hey everyone,
I need to find an integrating factor of the form x^n*y^m, to solve a differential equation i have... however i do not know the process to solve for an integration of this form.. .any help??
Thanks
Steph
 
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can you give an example problem that your working on?
 
the problem is ( 12 + 5xy )dx + (6 (x/y)+ 3x^2)dy =0
and it says, find an integrating factor of the form (x^n) * (y^m), and solve the equation...
thanks sweetie
steph
 
RULE 1: Mathematics problems are not solved by staring at a problem until you remember the answer! They are solved by plugging things in and doing the algebra.
So TRY!

If you multiply the equation by x^ny^m you get
(12x^ny^m+ 5x^{n+1}y^{m+1})dx+ (6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)dy= 0

In order for that to be an exact equation, you must have
(12x^ny^m+ 5x^{n+1}y^{m+1})_y= (6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)_x

Do the derivatives and see what m and n must be for those to be equal!

(12x^ny^m+ 5x^{n+1}y^{m+1})_y= 12mx^ny^{m-1}+ 5(m+1)x^{n+1}y^m
(6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)_x= 6(n+1)x^ny^{m-1}+3(n+2)x^{n+1}y^m
Coefficients of the same powers must be equal. That gives two equations for m and n.
 
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